F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07BDF (DGBTRF)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07BDF (DGBTRF) computes the $LU$ factorization of a real $m$ by $n$ band matrix.

## 2  Specification

 SUBROUTINE F07BDF ( M, N, KL, KU, AB, LDAB, IPIV, INFO)
 INTEGER M, N, KL, KU, LDAB, IPIV(min(M,N)), INFO REAL (KIND=nag_wp) AB(LDAB,*)
The routine may be called by its LAPACK name dgbtrf.

## 3  Description

F07BDF (DGBTRF) forms the $LU$ factorization of a real $m$ by $n$ band matrix $A$ using partial pivoting, with row interchanges. Usually $m=n$, and then, if $A$ has ${k}_{l}$ nonzero subdiagonals and ${k}_{u}$ nonzero superdiagonals, the factorization has the form $A=PLU$, where $P$ is a permutation matrix, $L$ is a lower triangular matrix with unit diagonal elements and at most ${k}_{l}$ nonzero elements in each column, and $U$ is an upper triangular band matrix with ${k}_{l}+{k}_{u}$ superdiagonals.
Note that $L$ is not a band matrix, but the nonzero elements of $L$ can be stored in the same space as the subdiagonal elements of $A$. $U$ is a band matrix but with ${k}_{l}$ additional superdiagonals compared with $A$. These additional superdiagonals are created by the row interchanges.

## 4  References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5  Parameters

1:     $\mathrm{M}$ – INTEGERInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{M}}\ge 0$.
2:     $\mathrm{N}$ – INTEGERInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     $\mathrm{KL}$ – INTEGERInput
On entry: ${k}_{l}$, the number of subdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{KL}}\ge 0$.
4:     $\mathrm{KU}$ – INTEGERInput
On entry: ${k}_{u}$, the number of superdiagonals within the band of the matrix $A$.
Constraint: ${\mathbf{KU}}\ge 0$.
5:     $\mathrm{AB}\left({\mathbf{LDAB}},*\right)$ – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AB must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the $m$ by $n$ matrix $A$.
The matrix is stored in rows ${k}_{l}+1$ to $2{k}_{l}+{k}_{u}+1$; the first ${k}_{l}$ rows need not be set, more precisely, the element ${A}_{ij}$ must be stored in
 $ABkl+ku+1+i-jj=Aij for ​max1,j-ku≤i≤minm,j+kl.$
See Section 9 in F07BAF (DGBSV) for further details.
On exit: if ${\mathbf{INFO}}\ge {\mathbf{0}}$, AB is overwritten by details of the factorization.
The upper triangular band matrix $U$, with ${k}_{l}+{k}_{u}$ superdiagonals, is stored in rows $1$ to ${k}_{l}+{k}_{u}+1$ of the array, and the multipliers used to form the matrix $L$ are stored in rows ${k}_{l}+{k}_{u}+2$ to $2{k}_{l}+{k}_{u}+1$.
6:     $\mathrm{LDAB}$ – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07BDF (DGBTRF) is called.
Constraint: ${\mathbf{LDAB}}\ge 2×{\mathbf{KL}}+{\mathbf{KU}}+1$.
7:     $\mathrm{IPIV}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}},{\mathbf{N}}\right)\right)$ – INTEGER arrayOutput
On exit: the pivot indices that define the permutation matrix. At the $\mathit{i}$th step, if ${\mathbf{IPIV}}\left(\mathit{i}\right)>\mathit{i}$ then row $\mathit{i}$ of the matrix $A$ was interchanged with row ${\mathbf{IPIV}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$. ${\mathbf{IPIV}}\left(i\right)\le i$ indicates that, at the $i$th step, a row interchange was not required.
8:     $\mathrm{INFO}$ – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero. The factorization has been completed, but the factor $U$ is exactly singular, and division by zero will occur if it is used to solve a system of equations.

## 7  Accuracy

The computed factors $L$ and $U$ are the exact factors of a perturbed matrix $A+E$, where
 $E≤ckεPLU ,$
$c\left(k\right)$ is a modest linear function of $k={k}_{l}+{k}_{u}+1$, and $\epsilon$ is the machine precision. This assumes $k\ll \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$.

## 8  Parallelism and Performance

F07BDF (DGBTRF) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
F07BDF (DGBTRF) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The total number of floating-point operations varies between approximately $2n{k}_{l}\left({k}_{u}+1\right)$ and $2n{k}_{l}\left({k}_{l}+{k}_{u}+1\right)$, depending on the interchanges, assuming $m=n\gg {k}_{l}$ and $n\gg {k}_{u}$.
A call to F07BDF (DGBTRF) may be followed by calls to the routines:
• F07BEF (DGBTRS) to solve $AX=B$ or ${A}^{\mathrm{T}}X=B$;
• F07BGF (DGBCON) to estimate the condition number of $A$.
The complex analogue of this routine is F07BRF (ZGBTRF).

## 10  Example

This example computes the $LU$ factorization of the matrix $A$, where
 $A= -0.23 2.54 -3.66 0.00 -6.98 2.46 -2.73 -2.13 0.00 2.56 2.46 4.07 0.00 0.00 -4.78 -3.82 .$
Here $A$ is treated as a band matrix with one subdiagonal and two superdiagonals.

### 10.1  Program Text

Program Text (f07bdfe.f90)

### 10.2  Program Data

Program Data (f07bdfe.d)

### 10.3  Program Results

Program Results (f07bdfe.r)